Given the payoff matrix, we explore the evolutionary dynamics between learning rules using the replicator equation [1]}. The replicator equation describes deterministic evolution in infinite populations. Let \(x\in \left[0,1\right]\)  denote the frequency of FMTL in the population. The frequency of selfish learners is \(1-x\) . Using the payoffs of Eq. REF , the fitness values of the two types are
\(f_{\textrm {F}}\left(x\right) &= x \,a_{\textrm {FF}} + \left(1-x\right) a_{\textrm {FS}} ; \\f_{\textrm {S}}\left(x\right) &= x \,a_{\textrm {SF}} + \left(1-x\right) a_{\textrm {SS}} ,\) 
